随机面积：正方形 vs 矩形

Peter and Paula both want to cut out a rectangular piece of paper. Because they are both probabilists they determine the exact form of the rectangle by using realizations of a positive rv, say U, as follows. Peter is lazy and generates just a single realization of this rv; he then cuts out a square that has length and width equal to this value. Paula likes diversity and generates two independent realizations of U. She then cuts out a rectangle with width equal to the first realization and length equal to the second realization.

a. Will the areas cut out by Peter and Paula differ in expectation?

b. If they do, is Peter's or Paula's rectangle expected to be larger?